# Hybrid Modelling in Nuclear Fusion

# 1 - Background

It has long been known that the MHD model is limited in its ability to model plasma phenomena. The hybrid kinetic-MHD model captures kinetic effects that are not possible to simulate with fluid MHD simulation alone. The addition of kinetic particle effects captures the wave-particle interactions lost in the conventional MHD approximation. Wave-particle interactions are known to strongly affect various MHD instabilities such as internal kink stabilization, generation of sawtooth and fish bone instabilities, and toroidal Alfven eigenmodes.

# 2 - The Hybrid Kinetic - MHD Model

The hybrid kinetic-MHD equations are constructed by considering the three species consisting of an ions species, an electrons species, and a hot particle species and summing their governing equations:

where f is the phase space distribution function and n is the density, Vs the flow velocity, **P**s the stress tensor, **F**s the Lorentz force, and the subscript *s* denotes the species. Multiplying each continuity equation by species mass and summing, gives the mass continuity equation

Applying the same prescription for the momentum equations and shifting from the lab frame to the center of mass frame, we have

where we have rewritten the stress tensor, **P**s, as the pressure tensor, **p**s, and advective derivative. For our definition of the pressure (defined with respect to the single fluid center of mass velocity), this equation is exact.